Yuji Liu

Studies on boundary value problems of singular fractional differential equations with impulse effects

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 70, 3 - 108 (2015)

MSC: 34A08 Fractional differential equations
  26A33 Fractional derivatives and integrals
  39B99 Difference and functional equations
  45G10 Other nonlinear integral equations
  34B37 Boundary value problems with impulses
  34B15 Nonlinear boundary value problems
  34B16 Singular nonlinear boundary value problems

Abstract:   In this paper, we firstly prove existence and uniqueness of solutions of Cauchy problems for nonlinear fractional differential equations involving the Caputo fractional derivative, the Riemann-Liouville derivative, the Caputo type Hadamard derivative and the Riemann-Liouville type Hadamard fractional derivatives of order α ∈ [n − 1, n) respectively by using Picard iterative technique under some suitable assumptions. Meanwhile, we get the iterative approximation solutions of these kind of Cauchy problems. Secondly we obtain exact expression of piecewise continuous solutions of the linear fractional differential equations. These results provide new methods to convert an impulsive fractional differential equation to a fractional integral equation. Thirdly, four classes of boundary value problems for singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. We allow the nonlinearity p(t)f (t, x) in fractional differential equations to be singular at t = 0, 1. Finally, by establishing existence results on solvability of two class of impulsive boundary value problems of fractional differential equations, we make a comparison on impulsive boundary value problems for two kinds of fractional differential equations, one has a single starting point and the other one has multiple starting points. In order to avoid misleading the readers, a mistake in [Impulsive integral boundary value problems of the higher-order fractional differential equation with eigenvalue arguments, Adv. Differ. Equa. (2015) 2015: 382] is also corrected.

Keywords:   higher order singular fractional differential equation, iterative approximation solution, impulsive boundary value problem, Riemann-Liouville fractional derivative, Caputo fractional derivative, Riemann-Liouville type Hadamard fractional derivative, Caputo type Hadamard fractional derivative, fixed point theorem
Notes:   Supported by the National Natural Science Foundation of China (No: 11401111), the Natural Science Foundation of Guangdong province (No: S2011010001900) and the Foundation for High-level talents in Guangdong Higher Education Project.

Seite generiert am 20.10.2016,   16:21   Uhr